Last year, Company X paid out a total of $1,050,000 in salaries to its

Assume the \(21^{st}\) guy receives the lowest salary \(x\)
and everybody else receives \(x+\frac{20}{100}x\) salary
\(x\,+\,20*(1.2x)\,=\,$10,50,000\)
\(x\,+\,24x\,=\,$10,50,000\)
\(25x\,=\,$10,50,000\)
\(x\,=\,\frac{$10,50,000}{25}\,=\,42,000\)

Answer C

Let x be the minimum salary,

Therefore, the maximum salary would be x * 0.2 + x = 1.2x
To get the minimum value for x, we have to maximise the value of all the other salaries, therefore we make all the other 20 people earn 1.2x

Total Amount
So we have x + 20(1.2x) = 1,050,000
x+ 24x = 1,050,000
25x = 1,050,000
x = 42,000
Answer choice C

[quote="Bunuel"]Last year, Company X paid out a total of $1,050,000 in salaries to its 21 employees. If no employee earned a salary that is more than 20% greater than any other employee, what is the lowest possible salary that any one employee earned?

(A) $40,000
(B) $41,667
(C) $42,000
(D) $50,000
(E) $60,000

Employee 1 earned $x(say)
Employee 2 will not earn more than $1.2x
Therfore, to minimize the salary of any one employee, we need to maximize the salaries of the other 20 employees
(1.2x*20)+x=1,050,000
Solving for x=$42,000
Answer C

VERITAS PREP OFFICIAL SOLUTION:

Here ask yourself the following questions:

1) The numbers do not have to be integers.

2) Zero is theoretically possible (but probably constrained by the 20% difference restriction)

3) Numbers absolutely can repeat (which will be very important)

4) What’s your strategy? If you want the LOWEST possible single salary, then use your answer to #3 (they can repeat) and give the other 20 salaries the maximum. That way your calculation looks like:

x + 20(1.2x) = 1,050,000

Which breaks out to 25x = 1,050,000, and x = 42000. And notice how important the answer to #3 was – by knowing that numbers could repeat, you were able to quickly put together a smart strategy to minimize one single value.

The larger lesson is crucial here, though – these problems are often (but not always) fairly basic mathematically, but derive their difficulty from a situation that limits some options or allows for more than you’d think via integer restrictions, the possibility of zero, and the possibility of repeat values. Ask yourself these four questions, and your answer to the first three especially will maximize your efficiency on the strategic portion of the problem.

Back-solving.

Use answer D: if minimum is 50k then 20% greater than that is 60k. Calculate whether the total salary base matches with that: 60*20+50 = way too much.

Use answer A (B is ugly so let us leave it for now): 40 + 48*20 = 1000 -> too little

Let us test C (B is ugly): 42 + 42*1.2*20 = 1050. Bingo!

Since the sum of all the salaries is a fixed amount (which is given to be $1,050,000), the lowest possible salary is obtained when all the salaries besides the lowest one are maximized. Since the greatest possible difference between any two salaries in this company is 20%, all the salaries besides the lowest salary should be 20% more than the lowest salary.

We can let the lowest salary = x and the greatest = 1.2x and create the following equation:

x + 20(1.2x) = 1,050,000

x + 24x = 1,050,000

25x = 1,050,000

x = 42,000

Answer: C

Dear Tmoni26,
You've explained the logic very concisely, clearly & perfectly.
Made perfect sense & helped me understand in least possible words.
Thank you!

Regards,
Saumya

This is a min/max problem can be attacked several different ways.
If the goal is to find the minimum possible salary among the 21 employees then you want to maximize the salary of 20 employees and see how much is left for the one remaining person with the lowest salary.

Since no salary can be more than 20% greater than any other, you want to assign the low salary as x and the remaining 20 salaries as 1.2x (20% greater than x).

The total salaries would then be x + 20(1.2x) = 25x. Dividing $1,050,000 by 25 you see that the lowest salary would be $42,000
the correct answer is (C).

Alternatively:

I am still not sure why we are taking 20 with max salary and 1 with lowest salary. By any means the max difference can come even if we take 1 person with max and 20 with min.
But with this, the answer changes. Could someone please help me with this.

Hi asthagupta,
https://gmatclub.com/forum/a-certain-ci ... 76217.html

Please check the explanation of Bunuel in the above link(OG Question), you may raise further specific queries (if any).

Hi,
can some one help confirm this approach is correct ?

50k is average.
The salaries need to be within a 20% around 50k.
x<50<1.2x
Since we know x<50
1.2x<60

Since we know 1.2x>50
x>50/1.2

50/1.2<x<50<1.2x<60

x>41.667k
closest value 42k.

we want to MINIMIZE the amount that 1 employee can get.

we can do this by MAXIMIZING the amount the other 20 employees can receive.

"no employee earned a salary that was More Than 20% greater than any other employee"

from this statement, the most that each of the other employees can earn is 20% Greater Than the Minimum Employee

20% Greater Than ------> (100 + 20)% = 120% of ------> (120/100) = 6/5


if the employee who want to have earn the MINIMUM amount possible earns = T

Each of the 20 other employees can only earn a MAXIMUM amount of = (6/5)*T

And all of these 21 salaries must SUM to $1,050,000

(20 employees) * (6/5)T dollars each + T = $1,050,000

(20) (6/5)T + T = 1,050,000

(4)(6)T + T = 1,050,000

24T + T = 1,050,000

25T = 1,050,000

T = (1,050,000 / 25) = Minimum Possible Salary that any one employee can earn given the constraints =


$42,000

We need to find the min/max.

Let \(x\) = least amount
Let \(1.2x\) = most amount

Since there are 21 employees:

\(x + 20(1.2x) = 25x\)

\(25x = 1,050,000\)

\(x = 42,000\)

Answer is C.

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